CN109450458A - The method for determining linear block codes performance bound at three error probabilities based on condition - Google Patents
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- H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
- H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
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Abstract
The invention discloses a kind of methods for determining linear block codes performance bound at three error probabilities based on condition, specifically follow the steps below: step S1, by linear block codes code word it is modulated after become send signal, signal is sent by the way that after additive white Gaussian noise channel, last receiving end receives received vector；Step S2, probability density function is calculated；Step S3, design conditions are at three error probabilities；Step S4, optimal higher-dimension spherical radius is calculated；Step S5, the linear block codes performance bound based on condition at three error probabilities is obtained, and then judges the maximum-likelihood decoding frame error probability of linear block codes.The present invention can be effectively prevented from the calculating again of repeat region decoding error probability, efficiently reduce computing repeatedly for decoding error probability region, present invention determine that linear block codes performance bound effectively improve existing spherical boundary, so that linear block codes performance bound is become tight and reduce maximum-likelihood decoding frame error probability.
Description
Technical field
The invention belongs to digital communications and stored digital field, in particular to a kind of to be determined based on condition at three error probabilities
The method of linear block codes performance bound.
Background technique
In order to solve reliable efficient communication issue under complex scene, suitable for the linear block codes under all kinds of communication systems
It is constantly suggested, the quality for how investigating a Linear codes code performance is particularly important.The decoding error of linear block codes
Probability can effectively measure the quality of code, however, in most cases, the decoding error probability of linear block codes can not be used
One exact expression formula is described.Therefore, decoding error probability relies primarily on the Monte-Carlo Simulation and solution of computer
Bound is estimated.Traditional method is to be judged using computer Monte-Carlo Simulation, but this method is than relatively time-consuming
Energy consumption needs to do many times when being especially to higher-dimension code or high s/n ratio (signal-to-noise ratio, SNR) first
Experiment, therefore, computer must work without cessation many days.Secondly when the bit error rate is lower, there is also can not be imitative with Monte Carlo
The case where really obtaining bit decoding error probability.Again, the result of Monte-Carlo Simulation is difficult directviewing description and makes mistake probability value
Relationship between system parameter, thus can not be with regard to how to improve the guidance of advancing a theory property in system performance problems.For illiteracy
The shortcomings that special Caro emulates, the maximum-likelihood decoding error probability bound technology of error correcting code is more concerned.For upper bound technology
For, when calculating the upper bound frame error ratio (Frame-error rate), generally only depend on code weight spectrum (or truncation weight
Amount spectrum), do not need the specific internal structure for obtaining code.On the one hand ginseng that Upper bound of error probability technology can instruct coding and decoding to design
Number selection, on the other hand can also be to avoid time-consuming Simulation Evaluation.2002, Sason pointed out (Variations on the
Gallager bounds,connections,and applications[J].IEEE Transactions on
Information Theory, 2002,48 (12): 3029-3051), most of upper bounds are the Gallager based on 1961
One upper bound technology (Gallager ' s first bounding technique, GFBT) model obtains, it may be assumed that
Wherein, E represents error event, and Pr (E) represents the probability of error event generation,yIt is received vector,It is
Received vectoryFall in regionOuter probability,The arbitrary region sent around signaling point is represented, whenWhen, it receives
The certain error of end decoding, therefore,It can be from
It obtainsWhenWhen, using based on pair-wise error probability (pair-wise
Error probability, PEP) joint circle carry out the upper bound.1994, Herzberg and Poltyrev (Techniques
of bounding the probability of decoding error for block coded modulation
Structures [J] .IEEE Transactions on Information Theory, 1994,40 (3): 903-911.) it is logical
Cross select the region Gallager for one using the higher-dimension ball that the signaling point where transmission signal vector is the centre of sphere (by optimization ball
Radius obtains the upper bound most tight under the shape), propose spherical boundary (Sphere bound, SB).2018, Liu Jia et al.
It is detailed in the article of entitled " further investigation of Binary Linear Block Codes spherical shape circle " that ZhongKai Agriculture Engineering Academy journal is delivered
The spherical boundary for carefully demonstrating Kasami et al. proposition is equivalent to the spherical boundary SB that Herzberg and Poltyrev is proposed.2016
Year, Zhao et al. proposes the nested region Gallager(Sphere bound revisited:A
new simulation approach to performance evaluation of binary linear codes over
AWGN channels[C]//Proceeding of International Symposium on Turbo Codes and
Iterative Information Processing, Brest, France, 2016:330-334.) upper bound has been changed to it is as follows
Expression formula:
Wherein, fu(r) probability is representedA computable upper bound,Indicate regionBoundary
Face, g (r) indicate stochastic variableProbability density function.Zhao et al. proposes condition pair-wise error probability, and again
Spherical boundary is derived.Since existing spherical boundary is all based on pair-wise error probability, error probability area has been inevitably resulted in
Domain computes repeatedly.It is a main cause for causing spherical boundary not tight that error probability region, which computes repeatedly,.Current method is
Computing repeatedly for error probability region is avoided or reduced by reducing the number of code word or reducing the item number of joint circle, it is famous
Scholar Agrell points out (On the voronoi neighbor ratio for binary linear block codes, "
IEEE Transactions on Information Theory, 1998,44:3064-3072), it is wrong to calculate maximum-likelihood decoding
Accidentally the probability upper bound need to only utilize the local weight spectrum (sending codeword set composed by code word Voronoi neighbours) for sending code word
?.The solution of local weight spectrum needs the positional relationship of all code words of theory analysis by combinatorics.Currently, only seldom
Code can solve local weight spectrum, for example, stochastic linear code, Hamming code, part BCH code and Reed Muller code, this
Although method can make the upper bound become tight to a certain extent, the local weight spectrum for being to solve for code is that a nondeterministic polynomial is tired
Difficult (non-deterministic polynomial hard, NP-Hard) problem.
Summary of the invention
The purpose of the present invention is to provide a kind of sides for determining linear block codes performance bound at three error probabilities based on condition
Method, solve spherical boundary in the prior art it is not compact and there are the computing repeatedly of error probability region, existing computer cover it is special
Caro emulation mode compare time consumption and energy consumption especially to higher-dimension code or high s/n ratio when, computation complexity is high and works as the bit error rate
When lower, the problem of there is also the case where can not obtaining bit decoding error probability with Monte-Carlo Simulation.
The technical scheme adopted by the invention is that the side of linear block codes performance bound is determined at three error probabilities based on condition
Method specifically follows the steps below:
Step S1, the definition to signal and received vector is sent:
Linear block codes code word isct∈ { 0,1 }, 0≤t≤n-1,
It is Binary Linear Block Codes, k is message length, and n is the code length of linear block codes, is modulated by binary phase shift keying BPSK
After become send signals=(s0,s1,…st,…sn-1), wherein st=1-2ct, 0≤t≤n-1, transmitting terminal will send signals
=(s0,s1,…st,…sn-1) be sent in additive white Gaussian noise channel awgn channel, the received vector that receiving end receives
ForWherein,zInterchannel noise is represented as n I.i.d. random variables group
At vector, each stochastic variable obey mean value be 0, variance σ2Gaussian Profile；
Step S2, probability density function g (r) is calculated；
Step S3, design conditions are at three error probability p3(r,i,j)；
Step S4, optimal higher-dimension spherical radius r is calculated1；
Step S5, the linear block codes performance bound based on condition at three error probabilities is obtained
And then judge the maximum-likelihood decoding frame error probability of linear block codes.
Further, the step S2 is specifically followed the steps below:
All-zero code word is obtained by step S1c (0)=(0,0 ..., 0) become to send out after binary phase shift keying BPSK modulation
Send signal vectors (0)The nested region of=(1,1 ..., 1), the selection of linear block codes performance bound is with transmission signal vectors (0)For
The centre of sphere, r >=0 are that the n of radius ties up ball, i.e.,
The probability density function of n dimension radius of a ball r
Wherein, σ represents interchannel noisezIn each stochastic variable take the standard deviation in normal distribution, Wherein, t indicates any one real variable,Indicate plural numberReal part numerical value；
Linear block codes has geometrical homogenization, and geometrical homogenization refers to geometry distribution of all code words in Euclidean space
It is that symmetrically and evenly, according to maximum-likelihood criterion, sending error probability caused by any code word all can be equal, it is subsequently assumed that channel
Middle transmission is all-zero code word.
Further, the step S3 is specifically followed the steps below:
Select a fixed codewordc (1)As reference code word, it is assumed that fixed codewordc (1)Hamming weight be d1=WH(c (1)) >=1, wherein function WH(c) indicate linear block codes code wordcHamming weight, definition i be linear block codes code wordcWith it is complete
Zero code wordc (0)Between Hamming distance, i.e. i=WH(c-c (0))；Definition j is linear block codes code wordcAnd fixed codewordc (1)It
Between Hamming distance, i.e. j=WH(c-c (1))；DefinitionFor Binary Linear Block CodesTriangle enumeration function, wherein X is the first dummy variable, and Y is the second dummy variable, Triangle Spectrum Bi,j(c (1)) it is line
Property block code code wordcNumber, Triangle Spectrum Bi,j(c (1)) and fixed codewordc (1)Correlation is omitted and is fixed when context is clear
Code wordc (1), define Bi,jFor the Triangle Spectrum of Binary Linear Block Codes, wherein 0≤i, j≤n,s (1)It is fixed codewordc (1)
The transmission signal vector obtained after BPSK is modulated, definitionAssuming that p3(r,
I, j) it indicates in eventMaximum-likelihood decoding under occurrence condition is at three error probabilities, then
Wherein,The n n-dimensional sphere n that expression radius is r, f (y) represent received vectoryProbability density function, Pr { }
Indicate probability；
Received vectoryIt is evenly distributed on the n n-dimensional sphere n that radius is rCondition is at three error probabilities
As shown in Fig. 1 (b), on spherical surfaceCondition at three error probabilities as composed by the intersection of two higher-dimension spherical crowns
The surface area fraction of geometry body surface area, that is, dotted portion and higher-dimension ball obtains, that is,
Wherein, d1For fixed codewordc (1)Hamming weight,
Further, the step S4 is specifically followed the steps below:
fuIt (r) is conditional joint circle, i.e.,
Wherein, Bi,j(c (1)) it is Binary Linear Block CodesTriangle Spectrum, p2(r,d1) it is the pairs of mistake of condition
Probability；
Condition pair-wise error probability p2(r,d1) as shown in formula (7):
As shown in Fig. 1 (a), condition pair-wise error probability p2(r,d1) by a higher-dimension spherical crown, that is, dotted portion and higher-dimension ball
Surface area fraction obtain；
Due to p3(r, i, j) is the non-decreasing continuous function about n dimension radius of a ball r, and p3(0, i, j)=0 and p3(+∞,
I, j)=(π+θ)/2 π, wherein φ is the variable of an expression angle, and θ is vectorAnd vectorFormed folder
Angle, therefore, conditional joint circle fu(r) and about n the non-decreasing continuous function of radius of a ball r, and f are tieed upuAnd f (0)=0u(+∞)
>=1, meanwhile, fu(r) in sectionIn be strictly increasing, andExistence anduniquess one optimal
Higher-dimension spherical radius r1Meet
Wherein, i=WH(c-c (0)), j=WH(c-c (1)), p2(r1,d1) be radius be r1High n-dimensional sphere n on transmitting terminal
Send bec (0), receiving end mistake is decoded into fixed codewordc (1)Condition pair-wise error probability；p3(r1, i, j) and it is to be in radius
r1High n-dimensional sphere n on transmitting terminal send bec (0), receiving end mistake is decoded into fixed codewordc (1)With linear block codes code wordc
Condition at three error probabilities.
The invention has the advantages that the present invention research code word space regularity of distribution, prevents or reduces error probability region
Compute repeatedly, by studying any three code word space regularities of distribution, find out the repeat region that decoding is easy error, the present invention
It can be effectively prevented from the calculating again of repeat region decoding error probability, efficiently reduce the repetition meter in decoding error probability region
It calculates, the present invention can effectively improve existing spherical boundary, so that linear block codes performance bound is become tight and make maximum-likelihood decoding
Frame error probability is reduced.The present invention proposes that compact linear block codes performance bound can be efficiently used for judging all Linear codes
The quality of code performance is not influenced by any condition.
Detailed description of the invention
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below
There is attached drawing needed in technical description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this
Some embodiments of invention for those of ordinary skill in the art without creative efforts, can be with
It obtains other drawings based on these drawings.
Fig. 1 is the geometric graph of condition pair-wise error probability and condition of the invention at three error probabilities；
Fig. 2 is original spherical boundary and the Linear codes of 1 Hamming code maximum-likelihood decoding frame error probability of the embodiment of the present invention
The comparison curves of code performance circle；
Fig. 3 is the convolution coding block diagram applied in the embodiment of the present invention 2；
Fig. 4 is original spherical boundary and the Linear codes of 2 convolutional code maximum-likelihood decoding frame error probability of the embodiment of the present invention
The comparison curves of code performance circle.
Specific embodiment
Below in conjunction with the embodiment of the present invention, technical scheme in the embodiment of the invention is clearly and completely described,
Obviously, described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.Based in the present invention
Embodiment, every other embodiment obtained by those of ordinary skill in the art without making creative efforts, all
Belong to the scope of protection of the invention.
Embodiment 1
Selected Binary Linear Block Codes are [7,4] Hamming code, wherein the code length n=7 of linear block codes, Chief Information Officer
Degree is k=4.Select minimum weight for 3 fixed codewordc (1)For with reference to code word, that is, WH(c (1))=d1=3, for calculating triangle
Shape spectrum.By calculating, triangle enumeration function is B (X, Y)=Y3+X3+6X3X4+6X4Y3+X4Y7+X7X4, obtain Bi,j(c (1)),
Using the present invention is based on the linear block codes performance bound that condition is determined at three error probabilities, linear block codes performance bound is obtained:
(1) givenUnder, wherein EbIt is the power of signal, N0It is the power spectral density of noise, obtainsIt traverses r ∈ (0 ,+∞), acquiresWherein, n=7；
(2) conditional joint circleWherein, k=4, n=7, d1=
3, it traverses r ∈ (0 ,+∞), acquires
With
(3) byObtain optimal higher-dimension spherical radius r1；
By calculating, g (r), f are obtainedu(r)、r1, bring into formula (10), obtain linear block codes performance bound.
Calculated result is shown in Fig. 2, as it is clear from fig. 2 that linear block codes performance bound proposed by the present invention is more compact, it more can be effective
The quality of linear block codes is assessed on ground, and the present invention passes through any three code word space regularities of distribution of research first, calculates triangle
Shape enumeration function, next is found out the repeat region of maximum-likelihood decoding error, reduces computing repeatedly for error probability region, have
Effect ground avoids the calculating again of repeat region decoding error probability, and being finally reached makes the change of linear block codes performance bound tightly and most
The purpose of maximum-likelihood decoding frame error probability reduction.
Embodiment 2
Selected Binary Linear Block Codes are [204,100] convolutional code, and wherein code length is n=204, and message length is
K=100, encoder is as shown in figure 3,1 bit u of every input exports two bit v by encoder(0)v(1), code rate isEncoder is the equipment of four state, and generator polynomial is G (D)=[1+D2,1+D+D2], wherein G (D) is indicated
The generator polynomial of convolutional code, D indicate delay operator.Select minimum weight for 5 fixed codewordc (1)For with reference to code word, that is, WH
(c (1))=d1=5, for calculating Triangle Spectrum, when calculating Triangle Spectrum, only consider the path in the first error event.Benefit
With the present invention is based on the linear block codes performance bound that condition is determined at three error probabilities, linear block codes performance bound is obtained, specifically
Calculating process is as follows:
(1) givenUnder, wherein SNR is signal-to-noise ratio, EbIt is the power of signal, N0It is the power spectrum of noise
Density acquiresIt traverses r ∈ (0 ,+∞), acquiresWherein, n=204；
(2) conditional joint circleWherein, k=100, n=204,
d1=5, it traverses r ∈ (0 ,+∞), acquires
With
(3) byObtain optimal higher-dimension spherical radius r1；
By calculating, g (r), f are obtainedu(r)、r1, bring into formula (10), obtain linear block codes performance bound.
Calculated result is shown in Fig. 4, and as seen from Figure 4, the linear block codes performance bound in the present invention is more compact than original spherical boundary, this
Invention passes through any three code word space regularities of distribution of research first, calculates triangle enumeration function, next finds out maximum seemingly
So repeat region of decoding error, reduces computing repeatedly for error probability region, is effectively prevented from repeat region decoding error
The calculating again of probability, being finally reached makes linear block codes performance bound become tight and maximum-likelihood decoding frame error probability reduction
Purpose.
The spherical boundary upper bound that linear block codes performance bound ratio Zhao of the invention et al. is proposed is more compact, and the present invention is calculating
fu(r) when, by studying any three code word space regularities of distribution, the repeat region of decoding error is found out, weight is effectively prevented from
The calculating again of multiple region decoding error probability, proposes and determines linear block codes performance bound at three error probabilities based on condition
Method.Again, proposed by the present invention based on condition is exact mathematical expression at the linear block codes performance bound of three error probabilities
Formula can quickly obtain the upper bound of the maximum-likelihood decoding error probability of linear block codes by calculating, and same up-to-date style (10) is to coding
Design has theoretic directive function, solves the drawbacks of time consumption and energy consumption of computer Monte-Carlo Simulation: 1, studying certain
A little codes, for example, performance of the performance or code under the maximum-likelihood decodings such as RS code, Turbo code and LDPC code under high s/n ratio
When, the time complexity of Monte-Carlo Simulation can be very high, for example, the maximum-likelihood decoding of RS code is that a uncertainty is multinomial
Formula difficult problem, the method that computer Monte-Carlo Simulation can not be utilized.Even if 2, certain error correcting codes can be by Monte Carlo
Emulation carries out the judgement of quality, but Monte-Carlo Simulation consumes energy.3, the result of Monte-Carlo Simulation is difficult directviewing description error
The accidentally relationship between probability value and system parameter, thus can not be with regard to how to improve the finger of advancing a theory property in system performance problems
It leads.Therefore, the present invention can be quickly used for judging the quality of linear block codes.
Each embodiment in this specification is all made of relevant mode and describes, same and similar portion between each embodiment
Dividing may refer to each other, and each embodiment focuses on the differences from other embodiments.Especially for system reality
For applying example, since it is substantially similar to the method embodiment, so being described relatively simple, related place is referring to embodiment of the method
Part explanation.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the scope of the present invention.It is all
Any modification, equivalent replacement, improvement and so within the spirit and principles in the present invention, are all contained in protection scope of the present invention
It is interior.
Claims (4)
1. the method for determining linear block codes performance bound at three error probabilities based on condition, which is characterized in that specifically according to following
Step carries out:
Step S1, the definition to signal and received vector is sent:
Linear block codes code word isct∈ { 0,1 }, 0≤t≤n-1,Be two into
Producing linear block code, k are message length, and n is the code length of linear block codes, are become after binary phase shift keying BPSK modulation
Send signals=(s0,s1,…st,…sn-1), wherein st=1-2ct, 0≤t≤n-1, transmitting terminal will send signals=(s0,
s1,…st,…sn-1) be sent in additive white Gaussian noise channel awgn channel, the received vector that receiving end receives isWherein,zInterchannel noise is represented to form as n I.i.d. random variables
Vector, each stochastic variable obey mean value be 0, variance σ2Gaussian Profile；
Step S2, probability density function g (r) is calculated；
Step S3, design conditions are at three error probability p3(r,i,j)；
Step S4, optimal higher-dimension spherical radius r is calculated1；
Step S5, the linear block codes performance bound based on condition at three error probabilities is obtained
And then judge the maximum-likelihood decoding frame error probability of linear block codes.
2. the method according to claim 1 for determining linear block codes performance bound at three error probabilities based on condition, special
Sign is that the step S2 is specifically followed the steps below:
All-zero code word is obtained by step S1c (0)=(0,0 ..., 0) become to send letter after binary phase shift keying BPSK modulation
Number vectors (0)The nested region of=(1,1 ..., 1), the selection of linear block codes performance bound is with transmission signal vectors (0)For ball
The heart, r >=0 are that the n of radius ties up ball, i.e.,
The probability density function of n dimension radius of a ball r
Wherein, σ represents interchannel noisezIn each stochastic variable take the standard deviation in normal distribution, Wherein, t indicates any one real variable,Indicate plural numberReal part numerical value.
3. the method according to claim 2 for determining linear block codes performance bound at three error probabilities based on condition, special
Sign is that the step S3 is specifically followed the steps below:
Select a fixed codewordc (1)As reference code word, it is assumed that fixed codewordc (1)Hamming weight be d1=WH(c (1)) >=1,
Wherein, function WH(c) indicate linear block codes code wordcHamming weight, definition i be linear block codes code wordcWith all-zero code wordc (0)Between Hamming distance, i.e. i=WH(c-c (0))；Definition j is linear block codes code wordcAnd fixed codewordc (1)Between Hamming
Distance, i.e. j=WH(c-c (1))；DefinitionFor Binary Linear Block CodesThree
Angular enumeration function, wherein X is the first dummy variable, and Y is the second dummy variable, Triangle Spectrum Bi,j(c (1)) it is linear block codes code
WordcNumber, Triangle Spectrum Bi,j(c (1)) and fixed codewordc (1)Correlation omits fixed codeword when context is clearc (1), fixed
Adopted Bi,jFor the Triangle Spectrum of Binary Linear Block Codes, wherein 0≤i, j≤n,s (1)It is fixed codewordc (1)It is modulated by BPSK
The transmission signal vector obtained afterwards, definitionAssuming that p3(r, i, j) is indicated in thing
PartMaximum-likelihood decoding under occurrence condition is at three error probabilities, then
Wherein,The n n-dimensional sphere n that expression radius is r, f (y) represent received vectoryProbability density function, Pr { } indicates general
Rate；
Received vectoryIt is evenly distributed on the n n-dimensional sphere n that radius is rCondition is at three error probabilities
On spherical surfaceCondition at three error probabilities by two higher-dimension spherical crowns intersect composed by geometry body surface area, that is, dotted lines
The surface area fraction of part and higher-dimension ball obtains, that is,
Wherein, d1For fixed codewordc (1)Hamming weight,
4. the method according to claim 3 for determining linear block codes performance bound at three error probabilities based on condition, special
Sign is that the step S4 is specifically followed the steps below:
fuIt (r) is conditional joint circle, i.e.,
Wherein, Bi,j(c (1)) it is Binary Linear Block CodesTriangle Spectrum, p2(r,d1) it is that mistake is general in pairs for condition
Rate；
Condition pair-wise error probability p2(r,d1) it is shown below:
Condition pair-wise error probability p2(r,d1) obtained by the surface area fraction of a higher-dimension spherical crown, that is, dotted portion and higher-dimension ball；
Wherein, φ is the variable of an expression angle；
Existence anduniquess one optimal higher-dimension spherical radius r1Meet
Wherein, i=WH(c-c (0)), j=WH(c-c (1)), p2(r1,d1) be radius be r1High n-dimensional sphere n on transmitting terminal send
Bec (0), receiving end mistake is decoded into fixed codewordc (1)Condition pair-wise error probability；p3(r1, i, j) be radius be r1's
Transmitting terminal transmission is on high n-dimensional sphere nc (0), receiving end mistake is decoded into fixed codewordc (1)With linear block codes code wordcItem
Part is at three error probabilities.
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| CN106230555A (en) * | 2016-07-29 | 2016-12-14 | 西安电子科技大学 | The stages cycle redundancy check method of polarization code |
| CN106788459A (en) * | 2016-12-12 | 2017-05-31 | 天津大学 | A kind of method of the estimation LDPC code error probability of Fast Convergent |
| CN107517095A (en) * | 2017-08-11 | 2017-12-26 | 北京理工大学 | A kind of polarization code coding/decoding method of unequal piece-wise verification |
| CN108259135A (en) * | 2018-01-11 | 2018-07-06 | 上海交通大学 | The weak polarization code construction method of anti-atmospheric turbulance based on Gaussian approximation theory |
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